Lecture 5: Intrinsic
semiconductors-conductivity
Contents
1 Carrier concentration in intrinsic Si
1
2 Carrier concentration in intrinsic semiconductors
2
3 Fermi level in intrinsic semiconductors
3
4 Temperature effect on ni
5
1
Carrier concentration in intrinsic Si
Consider intrinsic Si at room temperature. Intrinsic refers to pure Si, an
ideal single crystal with impurity concentrations below ppb (parts per billion). There is an equilibrium concentration of electrons and holes at room
temperature, due to thermal excitation. This is the intrinsic carrier concentration (ni ) given by
ni =
p
Eg
Nc Nv exp(−
)
kB T
(1)
Nc and Nv , the effective density of states at the band edges, are dependent
on temperature and the effective mass of the electron and holes respectively.
For Si, m∗e is 1.08 me and m∗h is 0.60 me . From this the effective density of
states can be calculated using
2πm∗e kB T 3
)2
h2
2πm∗h kB T 3
Nv = 2(
)2
h2
Nc = 2(
1
(2)
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At room temperature the values of Nc and Nv for Si are 2.81 × 1025 m−3
and 1.16 × 1025 m−3 . These are the total number of available states at the
band edges for the electrons and holes to occupy. Hence, the intrinsic carrier
concentration at room temperature can be calculated using equation 1 with
the band gap value of 1.10 eV . This gives ni equal to 1016 m−3 or 1010 cm−3 .
Thus, pure Si at room temperature has 1010 cm−3 electrons in the conduction
band and a equal number of holes in the valence band. To place this number
in context, the number of Si atoms per unit volume can be calculated from
the density (2.3 g cm−3 ) and the atomic weight (28 g mol−1 ). This works
out to be 5 × 1022 atoms cm−3 . So there is approximately 1 electron/hole
for every 5 × 1011 Si atoms!
This is a very low concentration compared to metals, where the electron
concentration is the comparable in magnitude to the atomic density and
explains why semiconductors have a poor conductivity compared to metals.
The conductivity is given by
σi = ni e(µe + µh ) =
p
Nc Nv exp(−
Eg
) e (µe + µh )
kB T
(3)
For Si, µe and µh are 1350 cm2 V −1 s−1 and 450 cm2 V −1 s−1 respectively. So,
the room temperature conductivity is 3 × 10−6 Ω−1 cm−1 or a resistivity of 3.5
× 105 Ωcm. The room temperature conductivity of Cu is 16 × 10−7 Ωcm,
nearly 12 orders of magnitude lower than Si and the difference is directly
related to the difference in concentration of the charge carriers.
2
Carrier concentration in intrinsic semiconductors
Pure Si is a very poor conductor. It is rarely used in the intrinsic form,
except for specific optical sensor applications, at liquid N2 temperatures.
Here is it usually lightly doped with Li to control the carrier concentration.
For higher conductivity, using equation 3, we either need a material with a
higher mobility or a higher intrinsic carrier concentration (i.e. lower band
gap). Mobility and band gap values for 3 commonly used semiconductors are
listed in table 1.
Based on mobilities, GaAs would be the material to choose for higher
conductivity but GaAs also has a higher band gap so that the intrinsic carrier
concentration would be low. The carrier concentration and conductivity for
pure GaAs can be calculated in a manner similar to that done for Si. The
effective mass values for GaAs are m∗e = 0.067 me and m∗h = 0.50 me . Using
2
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Table 1: Carrier mobilities and band gaps for some common semiconductors
Material µe (cm2 V −1 s−1 ) µh (cm2 V −1 s−1 ) Eg (eV )
Ge
3900
1900
0.66
Si
1350
450
1.10
GaAs
8500
400
1.43
equation 2 the effective density of states can be calculated to be Nc = 4.3
× 1023 m−3 and Nv = 8.85 × 1023 m−3 . This gives an intrinsic carrier
concentration of 2.4 × 106 cm−3 and hence a conductivity of 3.4 × 10−9
Ω−1 cm−1 . A similar calculation can be done for Ge so that table 1 can be
updated to give table 2 that contains the intrinsic carrier concentration and
conductivity for the semiconductors listed in table 1.
It can be seen that while a higher mobility can lead to an increase in
Table 2: Carrier concentration and conductivity for the three semiconductors
listed in table 1
Material Eg (eV )
Ge
0.66
Si
1.10
GaAs
1.43
ni (cm−3 )
2.3 × 1013
1010
2.4 × 106
σ (Ω−1 cm−1 )
0.02
3 × 10−6
3.4 × 10−9
conductivity the dominant term is the carrier concentration and consequently
the band gap. Since the carrier concentration has an exponential dependence
on Eg lowering Eg will have the greatest effect on conductivity. Based on
table 2, Ge would be the semiconductor material of choice due to its high
conductivity and low band gap. In fact, the first transistors were made
using Ge. But later, Si has come to dominate the microelectronics industry.
The reason for this is conductivity in Si can be increased by adding specific
impurities to the material. This is also coupled to the fact that Si is the
second most abundant material on earth (27%) while Ge is only present in
the parts per million amount.
3
Fermi level in intrinsic semiconductors
Fermi level is defined as the energy level separating the filled states from the
empty states at 0 K. It is also the highest filled energy level in a metal.
This definition needs to be modified for a semiconductor since these have
an energy gap between the filled states (valence band) and empty states
(conduction band). The expression for the carrier concentration (n or p) in
3
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a semiconductor is given by
(Ec − EF )
]
kB T
(EF − Ev )
p = Nv exp[−
]
kB T
n = Nc exp[−
(4)
In an intrinsic semiconductor n =p = ni . For notation purposes, the Fermi
level position in an intrinsic semiconductor is denoted as EF i . Hence, using
equation 4 and rearranging, the Fermi energy can be written as
EF i = Ev +
1
Nc
1
Eg − kB T ln( )
2
2
Nv
(5)
Since, the effective density of states depend on the carrier effective mass,
using equation 2, equation 5 can be rewritten as
EF i
3
m∗e
1
= Ev + Eg − kB T ln( ∗ )
2
4
mh
(6)
For Si, the intrinsic Fermi level is calculated to be 0.54 eV above the valence
band edge (Ev ) at room temperature . Since the band gap is 1.10 eV this
puts EF i very close to the center of the gap. If Nc = Nv (m∗e = m∗h ) then the
intrinsic Fermi level will be at the center of the band gap at all temperatures.
A schematic of the energy bands in an intrinsic semiconductor is shown in
figure 1. This picture is true for any intrinsic semiconductor, with the Fermi
level always close to the center of the gap.
In metals the Fermi level separates the filled and empty states. This is
because metals have partially filled energy bands and the Fermi level lies
within the band. This definition cannot be used for semiconductors. Another
definition is that Fermi level defines the average energy required to remove
an electron from the material. This is called the work function, denoted
by φ, and is the energy difference between the Fermi level and the vacuum
level. This definition can also used for semiconductors, with the realization
that electrons in the CB and the holes in the VB are linked. So the Fermi
level, defines the average potential energy per electron for electrical work.
If an external electric field is applied the Fermi level in the material will no
longer be uniform. This will be important when considering junctions formed
between different semiconductors and between semiconductors and metals.
4
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Figure 1: Band picture of an intrinsic semiconductor showing the VB and
CB edge and location of the Fermi level (EF i ). The electron and hole concentrations are equal and the Fermi level is located close to the center of the
band gap. Adapted from Principles of Electronic Materials - S.O. Kasap.
4
Temperature effect on ni
The temperature effect on the carrier concentration can be understood by
using equation 1. Temperature enters both the pre-exponential and the exponential term. Increasing temperature has two effects
1. Increases the effective density of states at the band edges (Nc and Nv ).
2. Decreases the exponential portion of equation 1, since T is in the denominator.
The exponential terms dominates, as can be seen by a semilog plot of ni vs.
inverse temperature, as shown in figure 2 for Si. The slope of the plot gives
the band gap of the semiconductor. Changing the semiconductor material,
changes the band gap and also the effective density of states. For different
semiconductors the plot of ni vs. T is similar with different slopes, corresponding to different Eg values. Intrinsic carrier concentrations for Ge, Si,
and GaAs are compared in figure 3. The slopes correspond to their respective
band gaps, given in table 1. The slope for GaAs is steepest, since it has the
highest band gap.
5
ni vs. inverse T
10
20
10
15
10
10
10
5
10
0
10
−5
10
−10
Slope = −Eg/2kb
n
i
(cm−3)
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1
2
3
4
5
6
7
8
9
Inv T (K−1)
10
−3
x 10
Figure 2: ni vs. inverse T for Si in the range 100-1000 K. The plot is a
Eg
. The dominating temperature depenstraight line with slope given by − 2k
B
dent term is the exponential one even though the effective density of states
also increase with temperature. The plot was generated in MATLAB.
ni vs. inverse T
16
10
Ge
Si
GaAs
14
10
12
−3
ni (cm )
10
10
10
8
10
6
10
4
10
2
10
0
10
2
2.5
3
3.5
Inv T (K−1)
4
4.5
5
−3
x 10
Figure 3: ni vs. inverse T for Ge, Si, GaAs in the range 200-500 K. The
plots are similar with different slopes given by their respective band gaps.
GaAs has the lowest carrier concentration while Ge has the largest. The plot
was generated in MATLAB.
6
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Increase in concentration and/or mobility has the effect of increasing conductivity. While concentration increases with temperature mobility decreases
with temperature. But because of the exponential term in equation 3, the
dominating term is ni . Consider two temperatures, T1 and T2 with intrinsic
carrier concentrations, ni1 and ni2 . Using equations 1 and 2 the ratio of the
carrier concentration is given by
T1 3
Eg 1
1
ni1
= ( ) 2 exp[−
(
−
)]
ni2
T2
2kB T1
T2
(7)
If T1 > T2 then ni1 > ni2 .
For Si, when the temperature is doubled from room temperature (300 K) to
600 K, using equation 7, the change in carrier concentration is given by
n600K
i
= 1.17 × 105
300K
ni
(8)
This leads to an approximately 5 orders of magnitude increase in conductivity. Problem is that almost all devices need to operate at or near room
temperature, so that increasing temperature would not be a feasible option.
So another method must be found to increase and control the conductivity
in Si. This is achieved by doping.
7